Topological Spaces

Recall the concepts of open and closed intervals in the set of real numbers R. The open interval (0, 1) includes all real numbers between 0 and 1, except 0 and 1.

However, for either endpoint, an infinite sequence may be defined that converges to it. For example, the sequence 1/2, 1/4, . . ., 1/2ⁱ converges to 0 as i tends to infinity. This means that we can choose a point in (0, 1) within any small, positive distance from 0 or 1, but we cannot pick one exactly on the boundary of the interval. For a closed interval, such as [0, 1], the boundary points are included.

The notion of an open set lies at the heart of topology. The open set definition that will appear here is a substantial generalization of the concept of an open interval. The concept applies to a very general collection of subsets of some larger space. It is general enough to easily include any kind of configuration space that may be encountered in planning.

A set X is called a topological space if there is a collection of subsets of X called open sets for which the following axioms hold:

1. The union of a countable number of open sets is an open set.

2. The intersection of a finite number of open sets is an open set.

3. Both X and ∅ are open sets.

Note that in the first axiom, the union of an infinite number of open sets may be taken, and the result must remain an open set. Intersecting an infinite number of open sets, however, does not necessarily lead to an open set.

For the special case of X = R, the open sets include open intervals, as ex-pected. Many sets that are not intervals are open sets because taking unions and intersections of open intervals yields other open sets. For example, the set

[∞ i=1

 1 3ⁱ, 2

3ⁱ



, (4.1)

which is an infinite union of pairwise-disjoint intervals, is an open set.

Closed sets Open sets appear directly in the definition of a topological space.

It next seems that closed sets are needed. Suppose X is a topological space. A subset C ⊂ X is defined to be a closed set if and only if X \C is an open set. Thus, the complement of any open set is closed, and the complement of any closed set is open. Any closed interval, such as [0, 1], is a closed set because its complement, (−∞, 0) ∪ (1, ∞), is an open set. For another example, (0, 1) is an open set;

x₁

U

x3

x2

O2

O₁

Figure 4.1: An illustration of the boundary definition. Suppose X = R², and U is a subset as shown. Three kinds of points appear: 1) x1 is a boundary point, 2) x2

is an interior point, and 3) x3 is an exterior point. Both x1 and x2 are limit points of U .

therefore, R\ (0, 1) = (−∞, 0] ∪ [1, ∞) is a closed set. The use of “(” may seem wrong in the last expression, but “[” cannot be used because −∞ and ∞ do not belong to R. Thus, the use of “(” is just a notational quirk.

Are all subsets of X either closed or open? Although it appears that open sets and closed sets are opposites in some sense, the answer is no. For X = R, the interval [0, 2π) is neither open nor closed (consider its complement: [2π,∞) is closed, and (−∞, 0) is open). Note that for any topological space, X and ∅ are both open and closed!

Special points From the definitions and examples so far, it should seem that points on the “edge” or “border” of a set are important. There are several terms that capture where points are relative to the border. Let X be a topological space, and let U be any subset of X. Furthermore, let x be any point in X. The following terms capture the position of point x relative to U (see Figure 4.1):

• If there exists an open set O1 such that x∈ O1 and O1 ⊆ U, then x is called an interior point of U . The set of all interior points in U is called the interior of U and is denoted by int(U ).

• If there exists an open set O2 such that x ∈ O2 and O2 ⊆ X \ U, then x is called an exterior point with respect to U .

• If x is neither an interior point nor an exterior point, then it is called a boundary point of U . The set of all boundary points in X is called the boundary of U and is denoted by ∂U .

• All points in x ∈ X must be one of the three above; however, another term is often used, even though it is redundant given the other three. If x is either an interior point or a boundary point, then it is called a limit point (or accumulation point) of U . The set of all limit points of U is a closed set called the closure of U , and it is denoted by cl(U ). Note that cl(U ) = int(U )∪ ∂U.

For the case of X = R, the boundary points are the endpoints of intervals. For example, 0 and 1 are boundary points of intervals, (0, 1), [0, 1], [0, 0), and (0, 1].

Thus, U may or may not include its boundary points. All of the points in (0, 1)

are interior points, and all of the points in [0, 1] are limit points. The motivation of the name “limit point” comes from the fact that such a point might be the limit of an infinite sequence of points in U . For example, 0 is the limit point of the sequence generated by 1/2ⁱ for each i∈ N, the natural numbers.

There are several convenient consequences of the definitions. A closed set C contains the limit point of any sequence that is a subset of C. This implies that it contains all of its boundary points. The closure, cl, always results in a closed set because it adds all of the boundary points to the set. On the other hand, an open set contains none of its boundary points. These interpretations will come in handy when considering obstacles in the configuration space for motion planning.

Some examples The definition of a topological space is so general that an incredible variety of topological spaces can be constructed.

Example 4.1 (The Topology of Rⁿ) We should expect that X = Rⁿ for any integer n is a topological space. This requires characterizing the open sets. An open ball B(x, ρ) is the set of points in the interior of a sphere of radius ρ, centered at x. Thus,

B(x, ρ) ={x^′ ∈ Rⁿ| kx^′− xk < ρ}, (4.2) in which k · k denotes the Euclidean norm (or magnitude) of its argument. The open balls are open sets in Rⁿ. Furthermore, all other open sets can be expressed as a countable union of open balls.¹ For the case of R, this reduces to representing any open set as a union of intervals, which was done so far.

Even though it is possible to express open sets of Rⁿas unions of balls, we pre-fer to use other representations, with the understanding that one could revert to open balls if necessary. The primitives of Section 3.1 can be used to generate many interesting open and closed sets. For example, any algebraic primitive expressed in the form H ={x ∈ Rⁿ | f(x) ≤ 0} produces a closed set. Taking finite unions and intersections of these primitives will produce more closed sets. Therefore, all of the models from Sections 3.1.1 and 3.1.2 produce an obstacle region O that is a closed set. As mentioned in Section 3.1.2, sets constructed only from primitives

that use the < relation are open. 

Example 4.2 (Subspace Topology) A new topological space can easily be con-structed from a subset of a topological space. Let X be a topological space, and let Y ⊂ X be a subset. The subspace topology on Y is obtained by defining the open sets to be every subset of Y that can be represented as U∩ Y for some open set U ⊆ X. Thus, the open sets for Y are almost the same as for X, except that the points that do not lie in Y are trimmed away. New subspaces can be constructed by intersecting open sets of Rⁿ with a complicated region defined by semi-algebraic models. This leads to many interesting topological spaces, some of

1Such a collection of balls is often referred to as a basis.

which will appear later in this chapter. 

Example 4.3 (The Trivial Topology) For any set X, there is always one triv-ial example of a topological space that can be constructed from it. Declare that X and ∅ are the only open sets. Note that all of the axioms are satisfied. 

Example 4.4 (A Strange Topology) It is important to keep in mind the al-most absurd level of generality that is allowed by the definition of a topological space. A topological space can be defined for any set, as long as the declared open sets obey the axioms. Suppose a four-element set is defined as

X ={cat, dog, tree, house}. (4.3)

In addition to∅ and X, suppose that {cat} and {dog} are open sets. Using the axioms, {cat, dog} must also be an open set. Closed sets and boundary points can be derived for this topology once the open sets are defined. 

After the last example, it seems that topological spaces are so general that not much can be said about them. Most spaces that are considered in topology and analysis satisfy more axioms. For Rⁿ and any configuration spaces that arise in this book, the following is satisfied:

Hausdorff axiom: For any distinct x₁, x₂ ∈ X, there exist open sets O1 and O2 such that x1 ∈ O1, x2 ∈ O2, and O1∩ O2 =∅.

In other words, it is possible to separate x1 and x2 into nonoverlapping open sets. Think about how to do this for Rⁿby selecting small enough open balls. Any topological space X that satisfies the Hausdorff axiom is referred to as a Hausdorff space. Section 4.1.2 will introduce manifolds, which happen to be Hausdorff spaces and are general enough to capture the vast majority of configuration spaces that arise. We will have no need in this book to consider topological spaces that are not Hausdorff spaces.

Continuous functions A very simple definition of continuity exists for topo-logical spaces. It nicely generalizes the definition from standard calculus. Let f : X → Y denote a function between topological spaces X and Y . For any set B ⊆ Y , let the preimage of B be denoted and defined by

f⁻¹(B) ={x ∈ X | f(x) ∈ B}. (4.4) Note that this definition does not require f to have an inverse.

The function f is called continuous if f⁻¹(O) is an open set for every open set O⊆ Y . Analysis is greatly simplified by this definition of continuity. For example, to show that any composition of continuous functions is continuous requires only a one-line argument that the preimage of the preimage of any open set always yields

an open set. Compare this to the cumbersome classical proof that requires a mess of δ’s and ǫ’s. The notion is also so general that continuous functions can even be defined on the absurd topological space from Example 4.4.

Homeomorphism: Making a donut into a coffee cup You might have heard the expression that to a topologist, a donut and a coffee cup appear the same. In many branches of mathematics, it is important to define when two basic objects are equivalent. In graph theory (and group theory), this equivalence relation is called an isomorphism. In topology, the most basic equivalence is a homeomorphism, which allows spaces that appear quite different in most other subjects to be declared equivalent in topology. The surfaces of a donut and a coffee cup (with one handle) are considered equivalent because both have a single hole. This notion needs to be made more precise!

Suppose f : X → Y is a bijective (one-to-one and onto) function between topological spaces X and Y . Since f is bijective, the inverse f⁻¹ exists. If both f and f⁻¹ are continuous, then f is called a homeomorphism. Two topological spaces X and Y are said to be homeomorphic, denoted by X ∼= Y , if there exists a homeomorphism between them. This implies an equivalence relation on the set of topological spaces (verify that the reflexive, symmetric, and transitive properties are implied by the homeomorphism).

Example 4.5 (Interval Homeomorphisms) Any open interval of R is home-omorphic to any other open interval. For example, (0, 1) can be mapped to (0, 5) by the continuous mapping x 7→ 5x. Note that (0, 1) and (0, 5) are each being interpreted here as topological subspaces of R. This kind of homeomorphism can be generalized substantially using linear algebra. If a subset, X ⊂ Rⁿ, can be mapped to another, Y ⊂ Rⁿ, via a nonsingular linear transformation, then X and Y are homeomorphic. For example, the rigid-body transformations of the previ-ous chapter were examples of homeomorphisms applied to the robot. Thus, the topology of the robot does not change when it is translated or rotated. (In this example, note that the robot itself is the topological space. This will not be the case for the rest of the chapter.)

Be careful when mixing closed and open sets. The space [0, 1] is not homeomor-phic to (0, 1), and neither is homeomorhomeomor-phic to [0, 1). The endpoints cause trouble when trying to make a bijective, continuous function. Surprisingly, a bounded and unbounded set may be homeomorphic. A subset X of Rⁿis called bounded if there exists a ball B ⊂ Rⁿ such that X ⊂ B. The mapping x 7→ 1/x establishes that (0, 1) and (1,∞) are homeomorphic. The mapping x 7→ tan⁻¹(πx/2) establishes

that (−1, 1) and all of R are homeomorphic! 

Example 4.6 (Topological Graphs) Let X be a topological space. The pre-vious example can be extended nicely to make homeomorphisms look like graph

Figure 4.2: Even though the graphs are not isomorphic, the corresponding topo-logical spaces may be homeomorphic due to useless vertices. The example graphs map into R², and are all homeomorphic to a circle.

Figure 4.3: These topological graphs map into subsets of R² that are not homeo-morphic to each other.

isomorphisms. Let a topological graph² be a graph for which every vertex cor-responds to a point in X and every edge corcor-responds to a continuous, injective (one-to-one) function, τ : [0, 1] → X. The image of τ connects the points in X that correspond to the endpoints (vertices) of the edge. The images of different edge functions are not allowed to intersect, except at vertices. Recall from graph theory that two graphs, G1(V1, E1) and G2(V2, E2), are called isomorphic if there exists a bijective mapping, f : V1 → V² such that there is an edge between v1 and v₁^′ in G1, if and only if there exists an edge between f (v1) and f (v₁^′) in G2.

The bijective mapping used in the graph isomorphism can be extended to produce a homeomorphism. Each edge in E1 is mapped continuously to its cor-responding edge in E2. The mappings nicely coincide at the vertices. Now you should see that two topological graphs are homeomorphic if they are isomorphic under the standard definition from graph theory.³ What if the graphs are not isomorphic? There is still a chance that the topological graphs may be homeo-morphic, as shown in Figure 4.2. The problem is that there appear to be “useless”

vertices in the graph. By removing vertices of degree two that can be deleted without affecting the connectivity of the graph, the problem is fixed. In this case,

2In topology this is called a 1-complex [440].

3Technically, the images of the topological graphs, as subspaces of X, are homeomorphic, not the graphs themselves.

graphs that are not isomorphic produce topological graphs that are not homeomor-phic. This allows many distinct, interesting topological spaces to be constructed.

A few are shown in Figure 4.3. 

4.1.2 Manifolds

In motion planning, efforts are made to ensure that the resulting configuration space has nice properties that reflect the true structure of the space of transforma-tions. One important kind of topological space, which is general enough to include most of the configuration spaces considered in Part II, is called a manifold. Intu-itively, a manifold can be considered as a “nice” topological space that behaves at every point like our intuitive notion of a surface.

Manifold definition A topological space M ⊆ R^m is a manifold⁴ if for every x∈ M, an open set O ⊂ M exists such that: 1) x ∈ O, 2) O is homeomorphic to Rⁿ, and 3) n is fixed for all x ∈ M. The fixed n is referred to as the dimension of the manifold, M . The second condition is the most important. It states that in the vicinity of any point, x ∈ M, the space behaves just like it would in the vicinity of any point y ∈ Rⁿ; intuitively, the set of directions that one can move appears the same in either case. Several simple examples that may or may not be manifolds are shown in Figure 4.4.

One natural consequence of the definitions is that m≥ n. According to Whit-ney’s embedding theorem [450], m≤ 2n+1. In other words, R²ⁿ⁺¹is “big enough”

to hold any n-dimensional manifold.⁵ Technically, it is said that the n-dimensional manifold M is embedded in R^m, which means that an injective mapping exists from M to R^m (if it is not injective, then the topology of M could change).

As it stands, it is impossible for a manifold to include its boundary points because they are not contained in open sets. A manifold with boundary can be defined requiring that the neighborhood of each boundary point of M is homeo-morphic to a half-space of dimension n (which was defined for n = 2 and n = 3 in Section 3.1) and that the interior points must be homeomorphic to Rⁿ.

The presentation now turns to ways of constructing some manifolds that fre-quently appear in motion planning. It is important to keep in mind that two

4Manifolds that are not subsets of R^mmay also be defined. This requires that M is a Hausdorff space and is second countable, which means that there is a countable number of open sets from which any other open set can be constructed by taking a union of some of them. These conditions are automatically satisfied when assuming M ⊆ R^m; thus, it avoids these extra complications and is still general enough for our purposes. Some authors use the term manifold to refer to a smooth manifold. This requires the definition of a smooth structure, and the homeomorphism is replaced by diffeomorphism. This extra structure is not needed here but will be introduced when it is needed in Section 8.3.

5One variant of the theorem is that for smooth manifolds, R²ⁿ is sufficient. This bound is tight because RPⁿ(n-dimensional projective space, which will be introduced later in this section), cannot be embedded in R²ⁿ⁻¹.

Yes

No Yes

Yes

Yes No

Yes No

Figure 4.4: Some open subsets of R² that may or may not be manifolds. For the three that are not, the point that prevents them from being manifolds is indicated.

manifolds will be considered equivalent if they are homeomorphic (recall the donut and coffee cup).

Cartesian products There is a convenient way to construct new topological spaces from existing ones. Suppose that X and Y are topological spaces. The Cartesian product, X×Y , defines a new topological space as follows. Every x ∈ X and y∈ Y generates a point (x, y) in X × Y . Each open set in X × Y is formed by taking the Cartesian product of one open set from X and one from Y . Exactly one open set exists in X× Y for every pair of open sets that can be formed by taking one from X and one from Y . No other open sets appear in X× Y ; therefore, its open sets are automatically determined.

A familiar example of a Cartesian product is R× R, which is equivalent to R². In general, Rⁿ is equivalent to R× Rⁿ⁻¹. The Cartesian product can be taken over many spaces at once. For example, R× R × · · · × R = Rⁿ. In the coming text, many important manifolds will be constructed via Cartesian products.

1D manifolds The set R of reals is the most obvious example of a 1D manifold because R certainly looks like (via homeomorphism) R in the vicinity of every point. The range can be restricted to the unit interval to yield the manifold (0, 1) because they are homeomorphic (recall Example 4.5).

Another 1D manifold, which is not homeomorphic to (0, 1), is a circle, S¹. In this case R^m = R², and let

S¹ ={(x, y) ∈ R² | x²+ y² = 1}. (4.5) If you are thinking like a topologist, it should appear that this particular circle is not important because there are numerous ways to define manifolds that are homeomorphic to S¹. For any manifold that is homeomorphic to S¹, we will

sometimes say that the manifold is S¹, just represented in a different way. Also, S¹ will be called a circle, but this is meant only in the topological sense; it only

sometimes say that the manifold is S¹, just represented in a different way. Also, S¹ will be called a circle, but this is meant only in the topological sense; it only